I'm considering the rotation group, as described by $SO(3)$ (i.e the set of $3 \times 3$ orthogonal matrices $R$ of determinant 1, with real components) and $SU(2)$ (the set of $2 \times 2$ complex matrices $U$ of determinant 1). The elements of these two groups could be represented like this (the unit matrix is implied implicitely): \begin{align} R(\alpha, n_k) &= 1 \cos \alpha + \sum_{k = 1}^3 \lambda_k \, n_k \sin \alpha + n \, n^{\top} (1 - \cos \alpha) \quad \in SO(3), \tag{1} \\ U(\alpha, n_k) &= e^{i \vec{\sigma} \, \cdot \, \vec{n} \, \alpha / 2} = 1 \cos (\alpha/2) + i \sum_{k = 1}^3 \sigma_k \, n_k \sin (\alpha/2) \quad \in SU(2), \tag{2} \end{align} where $\lambda_k$ ($k = 1, 2, 3$) are three $3 \times 3$ skew-symetrical matrices that generate the rotations, and $\sigma_k$ are the three $2 \times 2$ Pauli matrices. The components $n_k$ are three real numbers defining the rotation axis. We could write these components as this: $$n_k = \{ \sin\vartheta \cos \varphi, \: \sin \vartheta \sin \varphi, \: \cos \vartheta \}. \tag{3}$$ Of course, $\alpha$ is the rotation angle.
Now, I define a scalar product in $SU(2)$ as a trace: $$\langle \, U_1, \, U_2 \rangle = \frac{1}{2} \, \mathrm{Tr}(U_1^{\dagger} \, U_2), \qquad U_1, \, U_2 \, \in \, SU(2).\tag{4}$$ Calculating the differential $dU$ gives us a metric on $SU(2)$ (I set $\chi \equiv \alpha / 2$ to simplify things): $$ds^2 = \frac{1}{2} \, \mathrm{Tr}(dU^{\dagger} \, dU) = d\chi^2 + \sin^2 \chi \, (d\vartheta^2 + \sin^2 \vartheta \, d\varphi^2).\tag{5}$$ This is a well known metric on $\mathcal{S}_3$, the 3-sphere. This is fine, since $SU(2)$ is a compact Lie group, which have the topology of that sphere.
But then I tried to do the same with $SO(3)$ (the calculations are very messy in this case). I got this: $$\frac{1}{8} \, \mathrm{Tr}(dR^{\top} \, dR) = d\chi^2 + \sin^2 \chi \, (d\vartheta^2 + \sin^2 \vartheta \, d\varphi^2).\tag{6}$$ The fraction $\frac{1}{8}$ on the left part puzzles me. Why this factor, instead of $\frac{1}{3}$? This trace of $3 \times 3$ matrices doesn't look like a scalar product. My interpretation of (6) is lacking. How can we define a proper scalar product and Riemanian metric on $SO(3)$ that would show the 3-sphere hidden in it?